Generalised Transfer Functions of Neural Networks

When artificial neural networks are used to make model nonlinear dynamical systems, the system structure which can be extremely useful for analysis and design, is buried within the network architecture. In this paper explicit expressions for the frequency response or generalised transfer functions of both feedforward and recurrent neural networks are derived in terms of the network weights. The derivation of the algorithm is established on the basis of the Taylor series expansion of the activation functions used in a particular neural network. This leads to a representation which is equivalent to the nonlinear recursive polynomial model and enables the derivation of the transfer functions to be based on the harmonic expansion method. By mapping the neural network into the frequency domain information about the structure of the underlying nonlinear system can be recovered. Numerical examples are included to demonstrate the application to the new algorithm. These examples show that the frequency response functions appear to be highly sensitive to the network topology and training and that the time domain properties fail to reveal deficiencies in the trained network structure.